More on graphic toposes

C AHIERS DE

TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

F. W ILLIAM L AWVERE More on graphic toposes

Cahiers de topologie et géométrie différentielle catégoriques, tome 32, n^o1 (1991), p. 5-10

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© Andrée C. Ehresmann et les auteurs, 1991, tous droits réservés.

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MORE ON GRAPHIC TOPOSES

by

^F. William LAWVERE

CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE

CATÉGORIQUES

VOL. XXX II -1 (1991)

RÉSUMÉ. Un mondfde est

graphique

s’il satisfait A l’i- dentité de

Schützenberger-Kimura

^{aba = ab . Dans cet}

article on continue l’itude des topos, ^dits topos gra-

phiques, engendr6s

par les

objets

ayant ^un mondfde d’-

endomorphismes

^{de ce} type. On dimontre que dans ^un

topos graphique,

^{n est un}

cogdn6rateur.

In

[Boulder AMS]

^and

[Iowa AMAST]

^I

began

^the

study

^{of a}

special

class of combinatorial toposes. ^The

general defining

property ^{of these}

special graphic

toposes ^{is that}

they

^are

generated by

^those

objects

^whose

endomorphism

^{monoids are}

finite and

satisfy

^the

Schützenberger-Kimura identity

aba ⁼

ab ;

^{since this includes all localic}

(and

^{even all}

"one-way"

^i.e. sites in which all

endomorphism

^{monoids are}

trivial) examples,

^to

emphasise

^the

special "graphic"

^fea-

tures, ^some results are stated for the case when a

single

such

object ^suffices,

ⁱⁿ

particular,

^{when the} topos ^is

hyper-

connected. Monoids

satisfying

^the

Schutzenberger-Kimura identity

will also be called

"graphic

monoids" for

short;

^the

finiteness

assumption implies

^that

they

^have

enough

^con-

stants. The other main consequence of the finiteness assump- tion is that the toposes ^under consideration are

actually presheaf toposes,

^{so that the}

projectivity

^{of the generators}

will be used in calculations without further comment.

The

goals ^of

^this

study

^{are two-fold: to} illustrate in a

"simple"

^{context some}

topos-theoretic

constructions which are

thus far difficult to calculate in

general,

^{and to arrive}

eventually,

^via

geometric realization,

^{at automatic}

display

of

pictures

^of hierarchical structures such as

"hypercard", library catalogues,

^etc.

(The

actions of

non-commuting

^idem-

potents ^{are relevant to} the latter

problem

^{in that x. a can}

be

interpreted

^{as "the} part ^{of x}

reflecting ^{a" ,}

^rather

than

merely

^"the part ^{of x in a" as} with commutative

intersection.

^{In this note I describe some} results of this

study

^{which were} obtained after the

Bangor meeting.

Among

^{the easy}

propositions

^{stated at the}

meeting

^which

emphasize

the very

special

^{nature of a}

graphic

^monoid

^{M ,}

are the facts that _every left ideal is

actually

^{a two-sided}

ideal and that the lattice of all these is stable with res-

pect ^{to the usual}

right

^{action on all} truth-values

(= right ideals);

^{moreover, there is a lattice}

isomorphism

between the two-sided ideals and the essential

subtoposes

^{of the} topos ^of all

applications (=

_{d e f}

right actions)

^of

^{M ,}

^{with X-S c X}

defining

the "S-dimensional skeleton" of any

X ,

^{where S} is a

given

left ideal of M . If the

Aufhebung

^{of S is}

defined to be the smallest T;2 S such that every X. S is ^a

T-sheaf,

^{then the}

Aufhebung

^of 0 is

T o -

^{the set of all}

constants of

M ,

while the

Aufhebung

^of

To

^is

T 1 =

^the

smallest left ideal which is connected as a

right

^M-set

(see [Iowa] Prop.1 ); X. T 1

^{has the same} components ^{as X}

itself, justifying picturing

^{it as the} "one-dimensional" ske- leton of X . The

putative picture

^{of a}

general

^{X is the}

interlocking

union of the

pictures

^of

singular figures

^{of the}

generating

^{form. A}

significant

three-dimensional

generating

form is the "taco" which

pictures

^the

eight-element graphic

monoid of all those endofunctors of

"any"

category ^{4 obtai-} ned

by considering

^all

composites

ⁱⁿ any twice

aufgehobene adjoint analysis

^{of 4}

such as for

example,

^{an essential}

analysis

^of

graphic

toposes

with _every S-skeleton a

T-sheaf,

^{and the} empty ^{M-set an} S-sheaf.

Since the set D of all

subtoposes

^{of a} topos ^{a is a}

co-Heyting algebra,

^{we can} consider it as a closed category with T ~ S

meaning

^{T 2 S and ~}

meaning

^{u . For a}

combinatorial

topos

^all

subtoposes

^{are essential and thus}

admit a skeleton

functor,

^{so we can define a covariant}

functor

x display o cocomplete

D-enriched

categories

by sending

^{X to the set X of all}

subobjects

^{of X}

(using

direct

image

^on

morphisms) ^with

^{the enrichment}

and in

particular

X(O,B) = dim(B) .

Here we called the functor

"display

^6tat

^{0" ,}

because al-

though

^it

specifies important

information about the desired

picture,

the machine may still ^not know how to draw the lat- ter, unless we

specify

^the

pictures

^{of the}

generating forms;

the

examples

^{worked out in}

[Iowa AMAST]

show that these basic

pictures

may be

triangles,

but may also be squares ^or loz- enges, that

they

may be

closed,

but may also be

partly

open,

as for

example

^the "taco" which has two two-dimensional closed

sides,

but is open ^on top.

One

example

where the

display

is well understood is the topos of reflexive directed

graphs,

^{which are the}

applica-

tions of the three-element

graphic

^monoid A 1 ^{with two con-}

stants whose

generating

^{form is} h

-&#x3E;.

^. Since this mon-

oid may be considered ^to be the

theory

^of

graphs,

^a

precise justification

^{for the term}

"graphic"

^{in the more}

general

^case

is

provided by

^the

following

Proposition

^1. Within the category

of

^all

monoids,

^{the smal-} lest

equational variety containing

^the

single

three-element

example

e 1 ^is the category

of

all monoids

satisfying

^iden-

tically

^{aba = ab .}

Proof It suffices to show that for each _{n ,} the free mon-

oid

Fn

^{in the}

subcategory satisfying

^the

identity

^{can be}

embedded in a

product

^of

copies

^of e 1 ,

Fn

-&#x3E;

A1 In

^as

monoids,

^where

In

^{is a set.} The canonical choice for

In

is

In = (Fn,A1)

^{the set} of monoid

homomorphisms

^which

by

freeness is A1 ^{as a set,} for then there is a canonical hom-

omorphism Fn

-&#x3E;

A1 A1 given by double-dualization,

^{which to}

each

generating

letter of

Fn assigns

^the

corresponding

pro-

jection n

-&#x3E; Ai . ^{It can} be shown that this canonical hom-

omorphism

^is

injective,

^{i.e. that if u * w in}

Fn ,

^then

there exists an x E

An1

such that

u(x) * w(x)

ⁱⁿ A 1 . Here we use the fact that the elements _u,w are words with-

out

repetitions

^{in the n}

letters,

^{and that}

w(x) =

^the

first of the two constants of A1 ^{which occurs in the list}

x ^o w .

Note that while maps of

applications in

^{a fixed}

graphic

topos ^never increase dimension

(as

^is

expressed

in the funct-

orality

^of

display o above), by

^contrast maps between

graph-

ics often

do;

^for

example,

the "taco" is a

homomorphic image

of

F6 ,

but any free

graphic

^{monoid is one-} dimensional.

Although

every topos is the fixed

part

^{of a}

graphic

top-

os

(indeed

^one of the form

A 1 _/X)

for some left-exact idem- potent ^{functor on the}

latter,

^the

graphic

toposes ^{are them-} selves

special

^{in many} respects. This is born out

by

^the

following proposition

^which

generalizes

^{a remark made}

by

Borceux many years ago. Recall that in any topos, ^{n is an} internal cogenerator ^{in the}

x sense

^{that any}

^{object injects}

into some function space fix ;

however,

^to get ^{an actual} cogenerator ^{we must be able to}

replace

^the exponent

by

^{a dis-}

crete one, and this

usually

^{does not}

happen.

Proposition

^2.

(Gustavo Arenas)

^In any

graphic

topos, ⁿ is

a cogenerator.

Proof First note that

if A-&#x3E;X

can be

distinguished by

any map X ---7 n, then

they

^{can be}

distinguished by (the

characteristic map

of)

^some

principal subobject (image

^of

some

singular figure

B-&#x3E;

X) .

^{In the}

graphic

case we can even take B ⁼

A ;

for if the

endomorphisms

^{of A}

satisfy

the

graphic identity

^and x1,x2 ^{are two}

figures

^{in X of}

form

A ,

consider the characteristic map

[z]

^{of the}

image

of z ⁼ x 1 ^{or z =} X2 - If

[z] x1

⁼

[z]X2

^for both these

z , then there exist

endomorphisms ^a,b

^{so that both}

which

implies

The

computation

^{of the}

Aufhebung

^{relation on essential}

subtoposes

^{has not} yet been carried out in many

important examples

^{such as}

algebraic

^or differential

geometry.

^Even for

graphic

toposes ^this

computation

may be difficult. How- ever, in the latter case it can at least be reduced to a

question involving

congruence relations ^on ideals of the mon-

oid itself :

Proposition

^3.

If

^{S g T are}

left

ideals in a

graphic

^mon-

oid

M ,

then in the associated

graphic

topos

,

^one

has the condition

"every

^{X. S is a}

T-sheaf’ if

^and

only if for

every congruence relation ^{= on T as a}

right

^{M-set one}

has the

implication

Construing

the elements of the lattice ^D of essential

subtoposes

^{as refined}

dimensions,

^one would also like to add them. Since the infima in D do not

always

agree

[Kelly- Lawvere]

^with

intersections,

^it may be necessary in

general

to

replace

^{D with}

(2D)OP

^to get ^{a workable}

theory.

^The

idea is that the ^sum

S1*S2

^{of two} dimensions should be the smallest T for which

holds for the skeleta of all

X,Y

^{in the} topos.

(It

^{is usu-}

ally only

for dimension

0 ,

^{T =}

To ,

^that

() -

^{· T} pre-

serves

products). ^However,

^for

graphics

the infima do _agree with intersections and moreover we have

Proposition

^4.

If S1,S2,T

^are

left

^{ideals in a}

graphic monoid,

^{then the} above relation holds

if

^and

only if

Moreover, if T,T’

both have this property relative ^to

S1,S2

^{then so does T n}

T’ ;

^{thus the} intersection

of

^all

such T

gives

^{a well-}

defined

"sum"

of S1*-S2 .

Proof The

necessity

of the condition follows from consider-

ing ^X,Y

^{to be}

principal right

^ideals.

Conversely,

^{if it}

holds and

X E X.S1, y E Y · S2

^{for some}

arbitrary

^M-

applications X,Y

^with Sl,S2

witnessing

^that

fact,

^i.e.

XS1 = ^X, yS2 = y , then

choosing

^{t as in the condition}

gives

that x,y&#x3E; is in

(X

^x

Y) · T

^because

If u E T’

(for

^some other such

T’)

^{also fixes a}

given

pair

S1,S2 that t e T

fixes,

^{then tu E T n T’ since} these are

ideals,

^and

obviously

showing

that T n T’ serves as well or better.

One has

To*S

^{= S for all}

^{S ,}

^but clarification is needed on the

Question.

^What is the relation between the "successor"

S. T 1

and the

Aufhebung

^{S’ of S for}

graphic toposes?

^Recall

that Michael Zaks

(unpublished)

^{showed for the}

simplicial

topos ^{that while these two} constructions

give equal

^results

for S =

To,T1,T2, by

^{contrast for S 2}

T 3

^{one has ins-}

tead that the doubled dimension S*S ⁼

S’*Ti

^{for S’ the}

(smallest) Aufhebung

^of

S ,

^i.e.

(since

dimensions reduce

to natural numbers

together

^{with ± oo in the}

simplicial case) that

^{n’ = 2n - 1} for n &#x3E; 3 .

REFERENCES

F.W.LAWVERE, Qualitative

distinctions between some toposes ^of

generalized graphs, Proceedings of

AMS Boulder 1987

Symposium

^on

Category Theory

^and

Computer Science, Contemporary

Mathematics 92

(1989)

^261-299.

G.M. KELLY &#x26; F.W.

LAWVERE,

^{On the,}

Complete

Lattice of Essential

Localizations,

^{Bull. Société}

Mathematique

^de

Belgique,

^XLI

(1989)

^289-319.

F.W.

LAWVERE, Display

^of

Graphics

and their

Applications,

^as

Exemplified by 2-Categories

^{and the}

Hegelian

"Taco",

Proceedings of

^{the First} International

Conference

^on

Algebraic Methodology

^and

Software Technology,

^The

University of Iowa, (1989)

^51-75.

(There

^{is an error in the version of the latter} paper distributed at the

Bangor Meeting:

^Free

graphic

^{monoids do}

not act

faithfully

^{on their}

constants.)

December

14, 1989,

^revised

August

^{7, 1990}

F. William Lawvere SUNY at Buffalo 106 Diefendorf Hall

Buffalo,

^{NY 14214}

USA